Prescribing discrete Gaussian curvature on polyhedral surfaces

TL;DR

This paper explores the problem of prescribing combinatorial alpha-curvature on polyhedral surfaces, extending discrete conformal theory and Kazdan-Warner type theorems to this setting, advancing the understanding of discrete curvature prescription.

Contribution

It proves Kazdan-Warner type theorems for combinatorial alpha-curvature on polyhedral surfaces using variational principles, generalizing prior results and supporting the conjecture of approximating smooth surfaces.

Findings

Established Kazdan-Warner type theorems for alpha-curvature

Extended discrete conformal theory to prescribe curvature

Provided a first step towards approximating smooth surfaces with polyhedral ones

Abstract

Vertex scaling of piecewise linear metrics on surfaces introduced by Luo is a straightforward discretization of smooth conformal structures on surfaces. Combinatorial $\alpha$-curvature for vertex scaling of piecewise linear metrics on surfaces is a discretization of Gaussian curvature on surfaces. In this paper, we investigate the prescribing combinatorial $\alpha$-curvature problem on polyhedral surfaces. Using Gu-Luo-Sun-Wu's discrete conformal theory for piecewise linear metrics on surfaces and variational principles with constraints, we prove some Kazdan-Warner type theorems for prescribing combinatorial $\alpha$-curvature problem, which generalize the results obtained by Gu-Luo-Sun-Wu on prescribing combinatorial curvatures on surfaces. Gu-Luo-Sun-Wu conjectured that one can prove Kazdan-Warner's theorems via approximating smooth surfaces by polyhedral surfaces. This paper takes the first step in this direction.